Critical, detection, and quantification levels

Let $C=[0,\mathrm{\infty })$ be a concentration parameter set and let $\mathsf{E}=((\mathbb{R},\mathcal{B}(\mathbb{R})),C,(P_c{)}_{c\in C})$ be a statistical experiment (Statistical experiments). The response $X(x)=x$ is a real random variable (Random elements and real random variables) with law $P_c$ at concentration $c$; $P_0$ is the blank law. Since each $(t,\mathrm{\infty })$ is open, it is Borel (The Borel sigma-algebra of a topological space), so the indicator rule below is measurable. For a threshold $t\in \mathbb{R}$, the positive-decision rule is ${\phi }_t(x)={\mathbf{1}}_{\{x>t\}}$. Its blank false-positive probability is $P_0(X>t)$, and its false-negative probability at $c>0$ is $P_c(X\le t)$. These are the null rejection probability and one minus the power for the simple null $\{0\}$ and point alternative $\{c\}$ (Randomized tests, Test size and power, The expectation of an indicator is the probability of the event, Basic identities for a probability measure).

For a target $\alpha \in (0,1)$, an exact critical level $L_C$ is a threshold satisfying $P_0(X>L_C)=\alpha \mathrm{.}$ A threshold with probability at most $\alpha$ is an $\alpha$-level threshold; equality need not be attainable for a discrete response. For a target $\beta \in (0,1)$, a detection limit $L_D$ is the least $c>0$ satisfying $P_c(X\le L_C)=\beta ,$ when such a least concentration exists. Thus $L_C$ is measured in response units, whereas $L_D$ is measured in concentration units.

When $0<E_c[X]<\mathrm{\infty }$ and ${\mathrm{Var}}_c(X)<\mathrm{\infty }$, the coefficient of variation is $\mathrm{CV}(c):=\frac{\sqrt{{\mathrm{Var}}_c(X)}}{E_c[X]}\mathrm{.}$ Expectation under $P_c$ is as in Expectation of a nonnegative or integrable random variable, and variance has the meaning in Moments, variance, and covariance on a probability space. For a declared tolerance $k_Q>0$, a quantification limit $L_Q$ is the least $c>0$ satisfying $\mathrm{CV}(c)\le k_Q$, when such a least concentration exists. This definition does not make the three limits universal: each is tied to the response law and the stated error or precision criterion.

The measurement procedure and its observation law must include any blank correction used in routine reporting. A below-threshold report is an event such as $X\le L_C$; its likelihood contribution is determined by the declared measurement law, not by substituting $L_C$ as though it were an observed response.